NURBS : Important and basic points

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Curve parameters

A NURBS curve can be created by just 3 parameters – degree, control point and knot vector.

• Knot vector: It is a sequence of some real numbers, in non-decreasing order. This sequence depends on the assumption of u values. If you are assuming that u varies from 0 to 1, then knot vector is the sequence of numbers, between 0 and 1. So basically knot values are u values. There is no significance of each knot value separately. It is the sequence, which is important. The total number of knots = number of control points + degree + 1. If the degree of the curve is 2 and there are 4 control points, then there will be total 7-knot values. A curve will be called as Non-Uniform, if the sequence of these knot values is non-uniform (e.g 0.0, 0.4, 0.5, 0.6, 0.9, 0.95, 1.0 ) and called as uniform if this sequence is uniform (e.g. 0.0, 0.167, 0.333, 0.5, 0667, 0.833, 1.0).
• Degree and order: It is the measure of the curvature of the curve. Order of the curve = degree + 1.
• Control points: The control points determine the shape of the curve. Each point is having 4 float values – x position, y position, z position and weight of the control point. The minimum number of control points required to render a NURBS curve is the same as the order of the curve. So you need at least 4 control points for degree 3 curve.
• U parameter: In mathematics, the equation of a curve can be expressed as a function of an independent variable, called as a parameter. In nurbs-calculator, we call this parameter as u parameter. U parameter is used to calculate the coordinates of a point, on the curve, from its parametric equation. Generally, it is assumed that u is 0 at the start of the curve and 1 at the end. So at the middle of the curve, the value of u will be 0.5. Let’s assume that there is a line starting at (1,1) and ending at (2,2). Also, assume that u=0 at start and u=1 at the end. If you are asked to find the coordinates of the point, which lies at the middle of the line, then in the mathematical term you will say that what should be coordinates of the point at u=0.5. You can provide this u value to the parametric equation of the line and calculate the coordinates at the middle point. You can also assume that u=1 at start and u=2 at the end or u=10 at start and u=100 at the end. It is your assumptions which defines the middle point of the curve, so for 0&lt; u &lt; 1, middle point is at u = 0.5, for 1 &lt; u &lt; 2, middle is at u = 1.5 and for 10 &lt; u &lt; 100 middle is at u=55.
• Knot multiplicity: Multiplicity means repetition of knot values. Multiplicity is a property of knots that refers to the number of control points associated to a knot. On a cubic curve (degree 3), a knot can have a multiplicity of 1, 2, or 3, but not more than 3. Similarly for degree 2 curve, the multiplicity can’t be more than 2. If knot values have multiplicity = degree + 1 at the start, then the curve will be clamped at the start, that means the curve touches the first control point. If the multiplicity &lt; (degree+ 1) at the start, then the curve will not touch the first control point and the curve will start at u = k[p], where p=degree. Same is applicable for the end knots. If the multiplicity at the end is &lt; (degree + 1), then the curve will not touch the last control point and curve will end at u=k[m-p], where m = (total number of knots – 1). That means, if knot vector is = {0, 0.111, 0.222, 0.333, 0.445, 0.556, 0.667, 0.778, 0.889, 1.0} for degree 2 curve, then the curve will start from u = 0.2, which is k[2] and end at u=0.778, which is k[7]. If the multiplicity &gt; (degree + 1), then the curve will split into disjoint parts and the control point will be unused.

Bézier curves, B-Splines and NURBS

• Bézier curves: The number of control points in Bézier curves is exactly equal to (degree + 1). That means for degree 2 curve, there will be 3 control points. If we add 1 control point, then the degree will also increase by 1. This is the main limitation of Bezier curves. Sometimes we want to add more control point at some portion of the curve (to precisely control the shape of that portion), but if the degree is increased, then it will affect the shape of the entire curve.
• Since the number of control points and degree are fixed, so according to the formula –> k = n+p+1, the number of knots will always be 2*(degree+1).
• It is always clamped at start and end. That means the curve is tangent at the first and last control point.
• Since it is clamped at start and end, so first degree+1 knot values will be 0 and last degree+1 knot values will be 1. That means the knot vector of Bezier curve doesn’t contain any value other than 0 and 1.
• It may be a rational curve, means all the control points can have different weights.
• B-Splines:
• Minimum number of control points = degree + 1. More control points can be added without increasing the degree
• Knots values can be uniform or non-uniform.
• The curve may or may not be clamped at endpoints. That means the curve may or may not touch the endpoints.
• It is always irrational, so all the control points should have equal weights, usually 1.
• NURBS:
• All the properties are same as b-splines, except that it considers the weight of the control points. So each control point can have different weight. That is why it is called as rational b-splines.